216 Prof. Gr. M. Minchin on the Magnetic 



With these values (54) gives as the equation of a line of 

 force 



£-1 



2 



to the second power of the small quantity or — 



As a verification, the curve (59) should be at right angles 

 to the curve ® = const. given by (43), i.e., if we calculate 



the value of m-¥~ for each curve, the one should be the 

 dm 



negative reciprocal of the other, as far as small quantities of 



the second order. This is found to be the case ; for, from 



(59) we have 



dm m . / T 1 c 2 \ m 2 . , f /T w . c 2 7 g* 4 \\ 

 — — = — sm<f>( L— 1 + -. — 5 )--5-oSm26-< (L — 2) 2 + -j— 2 — jo — -, fll 

 md(j> 2a r V 4??z 2 / 8a 2 r L 4m 2 48 m 4 J 



which is ;—£ calculated from the equation (5) = const. 



dm 



c 2 

 If we denote the ratio -r— % by X, the result (59) may be 

 *±m 



written 



iL-^(L-l + X).cos^ +I g{L(i+X) + i-X 



-(iL-l-X + |\ 2 ) cos 20} = const. . . (60) 



Variable Current-density. — Let the current-density now 

 be supposed to vary inversely as the distance from the axis 

 OV. If at any point P in space T is the vector potential due 

 to a system of circular currents all having the same axis OV, 

 and if % is the distance of P from this axis, the equation of a 

 line of force is, as has been shown, 



Yx= constant. 



And r is the resultant vector potential at P due to a system 

 of currents running in filaments through the wire, the density 



of the typical current filament through Q (fig. 1) being ~j~ . -; 

 therefore 



r=.^J;jgKi+**)K-aH}jfdSj ■ . (6i) 





